基于Woodbury非线性方法的迭代算法对比分析  

COMPARATIVE ANALYSIS OF THE ITERATIVE ALGORITHMS FOR NONLINEAR METHOD BASED ON THE WOODBURY FORMULA

作　　者：贾硕 李钢[1] 李宏男[1,2] JIA Shuo;LI Gang;LI Hong-nan(State Key Laboratory of Costal and Offshore Engineering,Dalian University of Technology,Dalian,Liaoning 116024,China;College of Civil Engineering,Shenyang Jianzhu University,Shenyang,Liaoning 110168,China)

机构地区：[1]大连理工大学海岸和近海工程国家重点实验室,辽宁大连116024 [2]沈阳建筑大学土木工程学院,辽宁沈阳110168

出　　处：《工程力学》2019年第8期16-29,58,共15页Engineering Mechanics

基　　金：国家自然科学基金项目(51878112);中央高校基本科研业务费专项资金项目(DUT17ZD220);大连市高层次人才创新支持计划项目(2017RD04,2015R044)

摘　　要：在结构局部非线性求解过程中,刚度矩阵仅部分元素发生改变,此时切线刚度矩阵可写成初始刚度矩阵与其低秩修正矩阵和的形式,每个增量步的位移响应可用数学中快速求矩阵逆的Woodbury公式高效求解,但通常情况下迭代计算在结构非线性分析中是不可避免的,因此迭代算法的计算性能也对分析效率有重要影响。该文以基于Woodbury非线性方法为基础,分别采用Newton-Raphson(N-R)法、修正牛顿法、3阶两点法、4阶两点法及三点法求解其非线性平衡方程,并对比分析5种迭代算法的计算性能。利用算法时间复杂度理论,得到了5种迭代算法求解基于Woodbury非线性方法平衡方程的时间复杂度分析模型,定量对比了5种迭代算法的计算效率。通过2个数值算例,从收敛速度、时间复杂度和误差等方面对比了各迭代算法的计算性能,分析了各算法适用的非线性问题。最后,计算了5种算法求解基于Woodbury非线性方法平衡方程的综合性能指标。The elements of the stiffness matrix are often partially changed in the solution of local nonlinear problems,in which the tangent stiffness matrix can be written as the sum of the initial stiffness matrix and its low rank perturbation matrix so that the displacement response in each incremental step can be efficiently solved by the Woodbury formula that is used to calculate the inverse matrix in mathematics.However,the iterative calculation is often unavoidable in the structural nonlinear analysis,and the performance of the nonlinear iterative algorithm also has a great impact on the efficiency of the structural nonlinear analysis.This paper studies the iterative solution of the nonlinear method based on the Woodbury formula.The Newton-Raphson(N-R)method,the modified Newton method,the two-point method with three convergence order,the two-point method with four convergence order,and the three-point method,are chosen to solve the equilibrium equations of the nonlinear method based on the Woodbury formula,and the performance of these five iterative algorithms is compared.The time complexity analysis models of the five iterative algorithms solving the equilibrium equations of the nonlinear method based on the Woodbury formula are obtained,and the efficiency of the five algorithms is quantitatively compared.The calculation performance of the five iterative algorithms is compared through two cases from the perspective of convergence rate,time complexity and error;then the applicable nonlinear problems of the five algorithms are analyzed.Finally,the comprehensive performance index of the five iterative algorithms solving the equilibrium equations of the nonlinear method based on the Woodbury formula is calculated.

关 键 词：Woodbury公式 Newton-Raphson法 修正牛顿法 两点法 三点法 收敛性 时间复杂度 

分 类 号：TU311.4[建筑科学—结构工程]